Simple groups,maximal subgroups,and probabilistic aspects of profinite groups

We show that a finite simple group has at mostn ^1.875+o(1) maximal subgroups of indexn. This enables us to characterise profinite groups which are generated with positive probability by boundedly many random elements. It turns out that these groups are exactly those having polynomial maximal subgroup growth. Related results are also established.     

A generalization of the fast LUP matrix decomposition algorithm and applications

We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(m^α?1n) time, where the complexity of matrix multiplication is O(m^α). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(m^α?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations $\text{A}\text{x}\text{=}\text{b}$ (where $\text{b}$ is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, $\text{A}{}^1$, of A (i.e., $\text{AA}{}^1\text{A = A}$). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.     

Size direction games over the real line. III

We prove, using the continuum hypothesis, thatD (the direction player) has a winning strategy in {ie442-1} for some uncountableX, and that there is an uncountableX which intersects each perfect nowhere-dense set of reals in a countable set such thatD does not win in {ie442-2} for everya. We also give another proof to the fact that Γ^S (X) is a win forD is countable.     

Reflection classes whose cubes cover the alternating group

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ? 2) of A. It was known that for no REC X, X² = Alt(n) holds, and that for some RECs X, X⁴ = Alt(n) holds (n ? 5). Let i > 0, and let c(θ) denote the number of cycles of θ?S(n). Let X_i = {ψ ∈ S(n): ψ² = 1, ψ has exactly i fixed points}. We prove that θ?X_i³ if and only if: (1) i ≡ n (mod 2); (2) The parity of X_i equals the parity of θ; and (3) $\text{i ?}\text{1}\text{3}\text{(n + 2 c(θ))}$. As a consequence, {X: X is a REC, X³ = Alt(n)} and {X: X is a REC, X³ = S(n) ? Alt(n)} are determined.     

Coherent electronic and nuclear dynamics for charge transfer in 1-ethyl-4-(carbomethoxy)pyridinium iodide

Although polaronic interactions and states abound in charge transfer processes and reactions, quantitative and separable determination of electronic and nuclear relaxation is still challenging. The present paper employs the amplitudes, polarizations, and phases of four-wave mixing signals to obtain unique dynamical information on relaxation processes following photoinduced charge transfer between iodide and 1-ethyl-4-(carbomethoxy)pyridinium ions. Pump-probe signal amplitudes reveal the coherent coupling of an underdamped 115 cm(-1) nuclear mode to the charge transfer excitation. Assignments of this recurrence to intramolecular vibrational modes of the acceptor and to modulation of the intermolecular donor-acceptor distance are discussed on the basis of a high-level density functional theory normal-mode analysis and previously observed wave packet dynamics of solvated molecular iodine. Nuclear relaxation of the acceptor induces sub-picosecond decay of the pump-probe polarization anisotropy from an initial value of 0.4 to an asymptotic value of -0.05. Electronic structure calculations suggest that relaxation along the torsional coordinate of the ethyl group is the origin of the anisotropy decay. Electric-field-resolved transient grating (EFR-TG) signal fields are obtained by spectral interferometry with a diffractive optic based interferometer. These measurements show that the signal phase and amplitude possess similar dynamics. Model calculations are used to demonstrate how the EFR-TG signal phase yields unique information on transient material resonances located outside the laser pulse spectrum. This effect can be rationalized in that the real and imaginary parts of the nonlinear polarization are related by the Kramers-Kronig transformation, which allows the dispersive component of the polarization response to exhibit spectral sensitivity over a larger frequency range than that defined by the absorption bandwidth.     

Rh-catalyzed enantioselective hydrogenation of vinyl boronates for the construction of secondary boronic esters

[reaction: see text] Rh-catalyzed hydrogenation of prochiral vinyl boronates occurs in an enantioselective fashion in the presence of the chiral ligand Walphos 1. This transformation provides access to chiral secondary organoboronates that are not available from alkene hydroboration reactions. The chiral reaction products should be useful in organic synthesis, and preliminary experiments suggest that they may participate in one-pot amination and homologation reactions.     

A forest building process on simple graphs

Consider the following process on a simple graph without isolated vertices: order the edges randomly and keep an edge if and only if it contains a vertex which is not contained in some preceding edge. The resulting set of edges forms a spanning forest of the graph.The probability of obtaining $k$ components in this process for complete bipartite graphs is determined as well as a formula for the expected number of components in any graph. A generic recurrence and some additional basic properties are discussed.